Discrete Convexity in Joint Winner Property

Transcription

1 Discrete Convexity in Joint Winner Property Yuni Iwamasa Kazuo Murota Stanislav Živný January 12, 2018 Abstract In this paper, we reveal a relation between joint winner property (JWP) in the field of valued constraint satisfaction problems (VCSPs) and M -convexity in the field of discrete convex analysis (DCA). We introduce the M -convex completion problem, and show that a function f satisfying the JWP is Z-free if and only if a certain function f associated with f is M -convex completable. This means that if a function is Z-free, then the function can be minimized in polynomial time via M -convex intersection algorithms. Furthermore we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values. Keywords: valued constraint satisfaction problems, discrete convex analysis, M- convexity 1 Introduction A valued constraint satisfaction problem (VCSP) is a general framework for discrete optimization (see [19] for details). Informally, the VCSP framework deals with the minimization problem of a function represented as the sum of small arity functions. It is known that various kinds of combinatorial optimization problems can be formulated in the VCSP framework. In general, the VCSP is NP-hard. An important line of research is to investigate which classes of instances are solvable in polynomial time, and why these classes ensure polynomial time solvability. Cooper Živný [2] showed that if a function represented as the sum of unary or binary functions satisfies the joint winner property (JWP), then the function can be minimized in polynomial time. This gives an example of a class of instances that are solvable in polynomial time. In this paper, we present the reason why JWP ensures polynomial time solvability via discrete convex analysis (DCA) [10], particularly, M -convexity [13]. DCA is a theory of convex functions on discrete structures, and M -convexity is one of the important convexity concepts in DCA. M -convexity appears in many areas such as operations research, economics, and game theory (see e.g., [10, 11, 12]). The results of this paper are summarized as follows: We reveal a relation between JWP and M -convexity. That is, we give a DCA interpretation of polynomial-time solvability of JWP. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, , Japan. yuni iwamasa@mist.i.u-tokyo.ac.jp Department of Business Administration, Tokyo Metropolitan University, Tokyo, , Japan. murota@tmu.ac.jp Department of Computer Science, University of Oxford, Oxford, OX1 3QD, United Kingdom. standa.zivny@cs.ox.ac.uk 1

2 To describe the connection of JWP and M -convexity, we introduce the M -convex completion problem, and give a characterization of M -convex completability. By utilizing a DCA interpretation of JWP, we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values. This study will hopefully be the first step towards fruitful interactions between VCSPs and DCA. Notations. Let R and R + denote the sets of reals and nonnegative reals, respectively. In this paper, functions can take the infinite value +, where a < +, a + = + for a R, and 0 (+ ) = 0. Let R := R {+ } and R + := R + {+ }. For a function f : {0, 1} n R, the effective domain is denoted as dom f := {x {0, 1} n f(x) < + }. For a positive integer k, we define [k] := {1, 2,..., k}. For x = (x 1, x 2,..., x n ) R n, we define supp + (x) := {i [n] x i > 0}. 2 Preliminaries Joint Winner Property. Let d i 2 be a positive integer and D i := [d i ] for i [r]. We consider a function f : D 1 D 2 D r R + represented as the sum of unary or binary functions as f(x 1, x 2,..., x r ) = c i (x i ) + c ij (x i, x j ), (1) i [r] 1 i<j r where c i : D i R + is a unary function for i [r] and c ij : D i D j R + is a binary function for 1 i < j r. Furthermore we assume c ij = c ji for distinct i, j [r]. A function f of the form (1) is said [2] to satisfy the joint winner property (JWP) if it holds that c ij (a, b) min{c jk (b, c), c ik (a, c)} (2) for all distinct i, j, k [r] and all a D i, b D j, c D k. A function f of the form (1) satisfying the JWP is said to be Z-free if it satisfies that argmin{c ij (a, c), c ij (a, d), c ij (b, c), c ij (b, d)} 2 (3) for any i, j [r] (i j), {a, b} D i (a b), and {c, d} D j (c d). Cooper Živný [2] showed that if f of the form (1) satisfies the JWP, then f can be minimized in polynomial time. In fact, they showed that if f satisfies the JWP, then f can be transformed into a certain Z-free function f in polynomial time such that a minimizer of f is also a minimizer of f. Moreover they showed that a Z-free function can be minimized in polynomial time. JWP appears in many contexts. For example, JWP identifies a tractable class of the MAX- 2SAT problem, which is a well-known NP-hard problem [6]. Indeed, for a 2-CNF formula ψ, we can represent the MAX-2SAT problem for ψ as a Boolean binary {0, 1}-valued VCSP instance. In this binary VCSP instance, JWP is equivalent to the following condition on ψ: if clauses (x 1 x 2 ) and (x 1 x 3 ) are contained in ψ, then so is (x 2 x 3 ). In addition, the AllDifferent constraint [15] and SoftAllDiff constraint [14] can be regarded as special cases of the JWP, and certain scheduling problems introduced in [1, 8] satisfy the JWP. See also Examples 5 8 in [2] for details. 2

3 M -Convexity. A function f : {0, 1} n R is said [10, 11] to be M -convex if for all x, y {0, 1} n and all i supp + (x y) there exists j supp + (y x) {0} such that f(x) + f(y) f(x χ i + χ j ) + f(y + χ i χ j ), (4) where χ i is the ith unit vector and χ 0 is the zero vector. A function f : {0, 1} n R is said [10] to be M 2 -convex if f can be represented as the sum of two M -convex functions. It is well known that M -convex functions can be minimized in polynomial time. Furthermore if we are given two M -convex functions g and h, we can minimize an M 2-convex function f = g + h in polynomial time by solving the so-called M -convex intersection problem. Theorem 1 ([16, Theorem 10][18, Theorem 6.5]; see also [12, Theorem 3.3]). A function f : {0, 1} n R with the zero vector in dom f is M -convex if and only if f satisfies the following two conditions: Condition 1: For all distinct i, j, k [n] and all z {0, 1} n with supp + (z) [n] \ {i, j, k}, it holds that f(z + χ i + χ j ) + f(z + χ k ) min{f(z + χ j + χ k ) + f(z + χ i ), f(z + χ i + χ k ) + f(z + χ j )}. (5) Condition 2: For all distinct i, j [n] and all z {0, 1} n with supp + (z) [n] \ {i, j}, it holds that f(z + χ i + χ j ) + f(z) f(z + χ i ) + f(z + χ j ). We pay special attention to quadratic M -convex functions. Using Theorem 1, we provide a necessary and sufficient condition for the M -convexity of a function f : {0, 1} n R of the form f(x 1, x 2,..., x n ) := h i x i + h ij x i x j ((x 1, x 2,..., x n ) {0, 1} n ), (6) i [n] 1 i<j n where we assume h ij = h ji and h i < + for i, j [n]. Lemma 2. A function f of the form (6) is M -convex if and only if it satisfies the following: h ij min{h ik, h jk } (i, j, k : distinct). h ij 0 (i, j : distinct). In Lemma 2, h ij can take the infinite value +, whereas all h ij s are assumed to be finite in the characterization in [7] and [11]. In particular, we refer to the first condition h ij min{h ik, h jk } (i, j, k : distinct) as the anti-ultrametric property. Note that no conditions are imposed on h i. The proof of Lemma 2 is in Section 5 By Lemma 2, we know that M -convexity of a function of the form (6) depends only on quadratic coefficients (h ij ) i,j [n]. We say that a function f of the form (6) is defined by (h ij ) i,j [n] if the quadratic coefficients of f is equal to (h ij ) i,j [n]. 3 M -Convexity in Joint Winner Property M -Convex Completion Problem. We introduce the M -convex completion problem, and give a characterization of an M -convex completable function on {0, 1} n defined by (h ij ) i,j [n]. The M -convex completion problem is the following: 3

4 Given: (h ij ) i,j [n] such that h ij R or h ij is undefined for every distinct i, j [n]. Question: By assigning appropriate values in R to undefined elements of (h ij ) i,j [n], can we construct an M -convex function f : {0, 1} n R of the form (6)? It should be clear that a defined element can be equal to + and the infinite value (+ ) may be assigned to undefined elements. If there is an appropriate assignment of (h ij ) i,j [n], then (h ij ) i,j [n] is said to be M -convex completable. If h ij < 0 or h ij < min{h jk, h ik } holds for some defined elements h ij, h jk, h ik, then we obviously know that (h ij ) i,j [n] is not M -convex completable. Hence in considering the M -convex completion problem, we assume that h ij 0, (7) h ij min{h jk, h ik } (8) for all defined elements h ij, h jk, h ik. For quadratic coefficients H := (h ij ) i,j [n] containing undefined elements, we define the assignment graph of H as a graph G H = ([n], E H ; w), where E H := {{i, j} i j and h ij is defined} and w : E H R + is defined by w({i, j}) := h ij for {i, j} E H. Then the following theorem holds. Theorem 3. H := (h ij ) i,j [n] is M -convex completable if and only if argmin e C w(e) 2 holds for every chordless cycle C of G H. The proof of Theorem 3 is in Section 5. Remark 4. Farach Kannan Warnow [4] introduced the matrix sandwich problem for ultrametric property, which contains the M -convex completion problem as a special case. They also constructed an O(m + n log n)-time algorithm for the matrix sandwich problem for ultrametric property, where m is the number of defined elements. In our setting, m = O(n 2 ). Hence, by using this algorithm, we can obtain an appropriate M -convex completion in O(n 2 ) time if one exists. An O(n 2 )-time algorithm based on Farach Kannan Warnow s algorithm is the following: Suppose that all h ij are finite (if there exists h ij with h ij = +, then we can redefine the value of h ij as a sufficiently large finite value M). Take any maximum forest F of G H. Let α 1 > α 2 > > α p be the distinct values of defined elements of (h ij ) {i,j} F. For k = 1,..., p 1, let F α k be the subgraph of F induced by the edges with weight at least α k, i.e., F α k := {{i, j} F h ij α k }. Then, for each {i, j} E H with i, j connected in G H, set h ij to α k, where k is the minimum number such that i, j is connected in F α k. For each {i, j} E H with i, j disconnected in G H, set h ij to α p. In this paper, we present a graphic characterization of M -convex completability. With this characterization, we provide a DCA interpretation of polynomial-time solvability of JWP. Transformation into a Function over {0, 1}. To connect JWP and M -convexity, we introduce a transformation of a function f : D 1 D 2 D r R into a function ˆf : {0, 1} U R, where U is the set of all assignments to variables, that is, U := {(1, 1), (1, 2),..., (1, d 1 ), (2, 1), (2, 2),..., (2, d 2 ),..., (r, 1), (r, 2),..., (r, d r )}. We consider the following correspondence between x = (x 1, x 2,..., x r ) D 1 D 2 D r and ˆx = (ˆx (1,1),..., ˆx (1,d1 ), ˆx (2,1),..., ˆx (2,d2 ),..., ˆx (r,1),..., ˆx (r,dr)) {0, 1} U : (1,x 1 ) (2,x 2 ) (r,x r) (x 1, x 2,..., x r ) (0,..., 0, ˇ1, 0,..., 0, 0,..., 0, ˇ1, 0,..., 0,..., 0,..., 0, ˇ1, 0,..., 0). }{{}}{{}}{{} d 1 d 2 d r (9) 4

5 That is, ˆx (i,a) = 1 means that we assign a to x i, and ˆx (i,a) = 0 means that we do not. In view of (9), define a function ˆf by { f(x) if there exists x satisfying (9), ˆf(ˆx) := (ˆx {0, 1} U ). + otherwise Note that minimizing f is equivalent to minimizing ˆf. Now we consider the transformation of f of the form (1) into ˆf, where f is given in terms of c i for i [r] and c ij for i, j [r]. We define f : {0, 1} U R + by f(ˆx) := c i (a)ˆx (i,a) + h (i,a),(j,b)ˆx (i,a)ˆx (j,b) (ˆx {0, 1} U ), (10) (i,a) U (i,a),(j,b) U, (i,a) (j,b) where h (i,a),(j,b) := { c ij (a, b) if i j, undefined if i = j. (11) We also define δ U : {0, 1} U R by { 0 if there exists x satisfying (9), δ U (ˆx) := + otherwise (ˆx {0, 1} U ), which is the indicator function for the feasible assignments. Then we have ˆf(ˆx) = f(ˆx) + δ U (ˆx) (ˆx {0, 1} U ), where arbitrary values in R may be assigned to the undefined elements h (i,a),(i,b) in f without affecting the value of ˆf. Indeed, if ˆx dom δu, then ˆx (i,a)ˆx (i,b) = 0 for all i [r] and all distinct a, b D i. Hence h (i,a),(i,b)ˆx (i,a)ˆx (i,b) = 0 holds for each undefined element h (i,a),(i,b) by the definition (11). In particular, the set of minimizers of both ˆf(ˆx) and f(ˆx) + δ U (ˆx) are the same. It is clear that δ U is M -convex (dom δ U is the base family of a partition matroid, which is a direct sum of matroids of rank 1). Hence if (h (i,a),(j,b) ) (i,a),(j,b) U has an M -convex completion ( h (i,a),(j,b) ) (i,a),(j,b) U, then f defined by ( h (i,a),(j,b) ) (i,a),(j,b) U is M -convex and ˆf = f + δ U is M 2 -convex. This means that ˆf can be minimized in polynomial time. We need the values of ( h (i,a),(j,b) ) (i,a),(j,b) U in a minimization algorithm of M 2-convex functions. A function of the form (1) satisfies the JWP if and only if h (i,a),(j,b) min{h (j,b),(k,c), h (i,a),(k,c) } holds for defined elements h (i,a),(j,b), h (j,b),(k,c), h (i,a),(k,c) given in (10). Hence (h (i,a),(j,b) ) (i,a),(j,b) U satisfies the assumptions (7) and (8) for the M -convex completion problem. Theorem 3 implies the following theorem (the proof is in Section 5). Theorem 5. For a function f of the form (1), let (h (i,a),(j,b) ) (i,a),(j,b) U be defined by (11). Then (h (i,a),(j,b) ) (i,a),(j,b) U is M -convex completable if and only if f (has the JWP and) is Z-free. 4 Algorithm By using a general algorithm for the M -convex intersection (minimization of M 2-convex functions), we can minimize Z-free functions of the form (1) in polynomial time. Suppose that we are given c i : D i R + for i [r], c ij : D i D j R + for 1 i < j r, and a Z-free function 5

6 f defined as (1). We can minimize f by minimizing ˆf = f + δ U with an M -convex intersection algorithm. Here we take advantage of the fact that all the vectors in dom δ U have a constant component sum, i.e., (i,a) U ˆx (i,a) = r for all ˆx dom δ U. This implies that δ U is an M-convex function [10] and we can use an M-convex intersection algorithm. An M-convex intersection algorithm is easier to describe than an M -convex intersection algorithm, though the time complexity is the same. Therefore we devise a minimization algorithm for Z-free functions via an M-convex intersection algorithm. Since the functions are defined on {0, 1} n, the proposed algorithm is actually a variant of valuated matroid intersection algorithms [9]. Specifically, let f r denote the restriction of f to the hyperplane containing dom δ U, i.e., f(ˆx) if ˆx (i,a) = r, f r (ˆx) := (i,a) U + otherwise. Then minimizing f + δ U is equivalent to minimizing f r + δ U, where f r and δ U are M-convex functions. The proposed algorithm consists of three steps. Step 1: On the basis of Theorem 5, we construct an M -convex function f : {0, 1} U R in (10) through an M -convex completion of (h (i,a),(j,b) ) (i,a),(j,b) U in (11). Step 2: We find a minimizer of f r, to be used as an initial solution in Step 3. Step 3: We find a minimizer of f r +δ U by the successive shortest path algorithm with potentials for the M-convex intersection [10] (see also [9, Section 5.2]). In Step 3 of the algorithm, we use the auxiliary graph Gˆx,ŷ = (V, Eˆx,ŷ ) defined for ˆx dom f r and ŷ dom δ U by V := {s, t} U, (12) Eˆx := {((i, a), (j, b)) (i, a), (j, b) U, ˆx + χ (j,b) χ (i,a) dom f r }, (13) Eŷ := {((i, a), (j, b)) (i, a), (j, b) U, ŷ + χ (i,a) χ (j,b) dom δ U }, (14) E + := {(s, (i, a)) (i, a) supp + (ˆx ŷ)}, (15) E := {((j, b), t) (j, b) supp + (ŷ ˆx)}, (16) Eˆx,ŷ := Eˆx Eŷ E + E (17) with the arc length function l = lˆx,ŷ : Eˆx,ŷ R given by { f r (ˆx + χ v χ u ) f r (ˆx) if (u, v) Eˆx, l(u, v) := 0 otherwise. (18) Note that, by the definition of δ U, we can also describe Eŷ as Eŷ = {((i, a), (i, b)) i [r], a, b D i, (i, a) supp + (ŷ), (i, b) supp + (ŷ)}. Algorithm for Z-free function minimization: Step 1: Find an M -convex completion ( h (i,a),(j,b) ) (i,a),(j,b) U of (h (i,a),(j,b) ) (i,a),(j,b) U, and define f : {0, 1} U R by f(ˆx) = c i (a)ˆx (i,a) + h (i,a),(j,b)ˆx (i,a)ˆx (j,b) (ˆx {0, 1} U ). (i,a) U (i,a),(j,b) U, (i,a) (j,b) 6

7 Step 2: Let ˆx {0, 1} U be the zero vector. While (i,a) U ˆx (i,a) < r, do the following: Step 2-1: Obtain (i, a) argmin{f(ˆx + χ (i,a) ) (i, a) U \ supp + (ˆx )}. Step 2-2: ˆx ˆx + χ (i,a). Step 3: Let p : V R be a potential defined by p(v) := 0 for v {s, t} U. ŷ dom δ U. While ˆx ŷ, do the following: Take any Step 3-1: Make the auxiliary graph Gˆx,ŷ. Define the modified arc length l p : Eˆx,ŷ R by l p (u, v) := l(u, v) + p(u) p(v) for (u, v) Eˆx,ŷ. Step 3-2: For each v V, compute the length p(v) of an s-v shortest path in Gˆx,ŷ with respect to the modified arc length l p. Let P be an s-t shortest path having the smallest number of arcs in Gˆx,ŷ with respect to the modified arc length l p. Step 3-3: For (i, a) U, ˆx (i,a) 1 if ((i, a), (j, b)) P Eˆx, ˆx (i,a) ˆx (i,a) + 1 if ((j, b), (i, a)) P Eˆx, ˆx (i,a) otherwise, ŷ(i,a) + 1 if ((i, a), (j, b)) P E ŷ, ŷ(i,a) ŷ(i,a) 1 if ((j, b), (i, a)) P E ŷ, otherwise. ŷ (i,a) For v V, p(v) p(v) + p(v). At the end of Step 2, we obtain a minimizer of f r. The validity of Step 2 is given in [13, Theorem 3.2]. The time complexity of this algorithm is as follows, where n := U = i [r] d i (the proof is in Section 5). Theorem 6. The proposed algorithm runs in O(nr 3 + nr log n + n 2 ) time. By improving the algorithm of running time O(n 3 ) given in [2], Cooper Živný [3] gave an O(n 2 log n log r)-time algorithm for minimizing Z-free functions of the form (1). Our proposed algorithm is faster than Cooper Živný s for some r (e.g., r = O(n1/3 )). Remark 7. In the VCSP framework, we assume that the function f of the form (1) is explicitly given. This means that the input size is proportional to d i + log c i (d) + log c ij (d, e), i [r] i [r] d D i 1 i<j r e D j and then the running time in Theorem 6 is strongly polynomial in the input size. On the other hand, if we assume that f is given by the value oracles for the functions c i and c ij, the input size of f is proportional to r + log d i + log c i (d) + log c ij (d, e). i [r] i [r] d D i 1 i<j r e D j In this case, the running time in Theorem 6 is pseudo-polynomial in the input size. d D i d D i 7

8 5 Proofs In this section, we give the proofs of Lemma 2, Theorem 3, Theorem 5, and Theorem 6. Proof of Lemma 2. (only-if part). Suppose that there exist distinct i, j, k [n] such that h ij < min{h jk, h ik }. Note that h ij < + holds. Then f(χ i + χ j ) + f(χ k ) = h i + h j + h k + h ij < h i + h j + h k + h jk = f(χ j + χ k ) + f(χ i ), f(χ i + χ j ) + f(χ k ) = h i + h j + h k + h ij < h i + h j + h k + h ik = f(χ i + χ k ) + f(χ j ) hold since h i, h j, h k, h ij < +. By Condition 1 of Theorem 1, f is not M -convex. Suppose that there exist distinct i, j [n] such that h ij < 0(< + ). Then f(χ i + χ j ) + f(χ 0 ) = h i + h j + h ij < h i + h j = f(χ i ) + f(χ j ) holds since h i, h j < +. By Condition 2 of Theorem 1, f is not M -convex. (if part). Take arbitrary distinct i, j, k [n] and z {0, 1} n with supp + (z) [n] \ {i, j, k}. If f(z + χ i + χ j ) = + or f(z + χ k ) = + holds, then Condition 1 of Theorem 1 obviously holds. We assume f(z + χ i + χ j ) < + and f(z + χ k ) < +. It holds that f(z + χ i + χ j ) = f(z) + h i + h j + h ip + h jp + h ij, (19) f(z + χ k ) = f(z) + h k + h kp. (20) Note that all terms appearing in (19) and (20) have finite values since f(z + χ i + χ j ) < + and f(z + χ k ) < + hold. Then we have Also we have f(z + χ i + χ j ) + f(z + χ k ) f(z + χ j + χ k ) + f(z + χ i ) 2f(z) + h i + h j + h k + h ip + h jp + 2f(z) + h j + h k + h i + h ij h jk. h jp + h kp + f(z + χ i + χ j ) + f(z + χ k ) f(z + χ i + χ k ) + f(z + χ j ) h ij h ik. By the assumption, it holds that h ij min{h jk, h ik }. Hence we obtain h kp + h ij h ip + h jk f(z + χ i + χ j ) + f(z + χ k ) min{f(z + χ j + χ k ) + f(z + χ i ), f(z + χ i + χ k ) + f(z + χ j )}. By the assumption of h ij 0, we also obtain for all distinct i, j [n]. f(z + χ i + χ j ) + f(z) f(z + χ i ) + f(z + χ j ) 8

9 Proof of Theorem 3. First we give a graphical interpretation for the anti-ultrametric property. For H := (h ij ) i,j [n] and α R, let us define EH α and V H α by Let G α H := (V H α, Eα H ). Then the following lemma holds: E α H := {{i, j} E H h ij α}, (21) V α H := {i e E α H such that i e}. (22) Lemma 8. H := (h ij ) i,j [n] satisfies the anti-ultrametric property if and only if each connected component of G α H is a complete graph for every α R. Proof. (only-if part). We show the contraposition. Suppose that for some α R there exists a non-complete graph among the connected components of G α H. Then there exist distinct i, j, k [n] with {i, j}, {j, k} EH α {i, k}. By the definition of Eα H, it holds that min{h ij, h jk } α > h ik. This means that {h ij, h jk, h ik } does not satisfy the anti-ultrametric property. (if part). Suppose that each connected component of G α H is a complete graph for all α R. To show the anti-ultrametric property of (h ij ) i,j [n], it suffices to prove h jk = h ik for all distinct i, j, k satisfying h ij > h jk. If h ik h ij, then there exists a non-complete graph among the connected components of G α H for α = h ij, which is a contradiction. If h ij > h ik > h jk, then there exists a non-complete graph among the connected components of G α H for α = h ik, which is a contradiction. If h jk > h ik, then there exists a non-complete graph among the connected components of G α H for α = h jk, which is a contradiction. Therefore we must have h jk = h ik. We are now ready to prove Theorem 3. Proof of Theorem 3. (only-if part). Suppose to the contrary that H := (h ij ) i,j [n] is M -convex completable and that there exists a chordless cycle C of G H with argmin e C w(e) = 1. Let C = {{i 1, i 2 }, {i 2, i 3 },..., {i m, i 1 }}, and consider the corresponding entries {h i1 i 2, h i2 i 3,..., h imi1 } of H. Note that h ipiq is undefined for p, q [m] with p q = 1 mod m. We may assume α := h i1 i 2 = min{h i1 i 2, h i2 i 3,..., h imi1 }. By the assumption of argmin e C w(e) = 1, we have min{h i2 i 3,..., h imi1 } > α. Since {h i1 i 2, h i2 i 3, h i1 i 3 } should satisfy the anti-ultrametric property, we have to assign α to h i1 i 3 to obtain an M -convex completion. Since {h i1 i 3, h i3 i 4, h i1 i 4 } should satisfy the anti-ultrametric property, we have to assign α to h i1 i 3 to obtain an M - convex completion. By repeating this procedure, we arrive at h i1 i m 1 = α. This is a contradiction, since h i1 i m 1 < min{h im 1 i m, h i1 i m } and hence the anti-ultrametric property fails for {h i1 i m 1, h im 1 i m, h imi1 }. (if part). For α R +, define SH α by S α H := {{i, j} E H i, j V α H, i and j are connected in G α H}. Let G α H := (V H α, Eα H Sα H ). Recall that Eα H and V H α are defined in (21) and (22). First we show that if each connected component of G α H is a complete graph for every α R +, then H is M -convex completable. Let α 1 > α 2 > > α p be the distinct values of defined elements of (h ij ) i,j [n] (α 1 can be the infinite value). We assign α 1 to each undefined element h ij such that {i, j} S α 1 H, α k to each h ij such that {i, j} S α k H \ Sα k 1 H for k = 2,..., p 1, and α p to each h ij such that {i, j} S α p 1 H. Then we obtain a certain completion H := ( h ij ) i,j [n] of (h ij ) i,j [n]. It is clear that each connected component of G α H is a complete graph for every α R. By Lemma 8, H satisfies the anti-ultrametric property. This means that H is M -convex completable. Next we show that if argmin e C w(e) 2 holds for every chordless cycle C of G H, then each connected component of G α H is a complete graph for every α R +. Take arbitrary α R + 9

10 and i and j which are connected in G α H (Note that vertex sets of connected components of G α H are the same as those of G α H ). It suffices to prove that {i, j} Eα H or {i, j} Sα H holds. Suppose to the contrary that there exist i and j such that {i, j} EH α and {i, j} Sα H hold. Let I be the set of such {i, j}. Let {i 0, j 0 } I be a pair of vertices such that the number of edges of a shortest i 0 -j 0 path on G α H is minimum in I. Since i 0 and j 0 are connected in G α H and {i 0, j 0 } SH α, we have {i 0, j 0 } E H. Moreover since {i 0, j 0 } EH α, h i 0 j 0 < α holds. Take a i 0 -j 0 shortest path P 0. Then P 0 {i 0, j 0 } is a chordless cycle of G H. Indeed, if P 0 {i 0, j 0 } has a chord in G H, there exist i and j satisfying {i, j } {i 0, j 0 } in P 0 {i 0, j 0 } such that EH α {i, j } E H by the minimality of I. Then {i, j } I and the number of edges of a shortest i -j path is smaller that those of P 0. However this is a contradiction to the minimality of {i 0, j 0 }. Hence P 0 {i 0, j 0 } is a chordless cycle of G H. It holds that h ij α for {i, j} P 0 and h i0 j 0 < α. Therefore we obtain argmin e P0 {i 0,j 0 } w(e) = 1. This contracts the assumption of argmin e C w(e) 2. Hence we have {i, j} EH α or {i, j} Sα H. Proof of Theorem 5. Let H := (h (i,a),(j,b) ) (i,a),(j,b) U, where the entries h (i,a),(j,b) with i = j are undefined. Recall that G H = (U, E H ; w) is the assignment graph of H. By the definition of E H and h (i,a),(j,b) in (11), we have E H = {{(i, a), (j, b)} i j, a D i, b D j }. By Theorem 3, H is M -convex completable if and only if every chordless cycle C satisfies the condition argmin e C w(e) 2. First we show that chordless cycles in G H have length 3 or 4. Take any chordless cycle C = {{(i 1, a 1 ), (i 2, a 2 )}, {(i 2, a 2 ), (i 3, a 3 )},..., {(i k, a k ), (i 1, a 1 )}} of G H. Since C is chordless, we have i 1 = i p for 3 p k 1 and i 2 = i q for 4 q k. This implies k 4, since otherwise we obtain i 1 = i 4 = i 2, contradicting the existence of an edge between (i 1, a 1 ) and (i 2, a 2 ). For a (chordless) cycle of length 3, say, C = {{(i 1, a 1 ), (i 2, a 2 )}, {(i 2, a 2 ), (i 3, a 3 )}, {(i 3, a 3 ), (i 1, a 1 )}} with i 1 i 2 i 3 i 1, the condition argmin e C w(e) 2 is equivalent to (2) for JWP. For a chordless cycle of length 4, say, C = {{(i 1, a 1 ), (i 2, a 2 )}, {(i 2, a 2 ), (i 3, a 3 )}, {(i 3, a 3 ), (i 4, a 4 )}, {(i 4, a 4 ), (i 1, a 1 )}} we have i 1 i 2, i 3 i 4, i 1 = i 3, i 2 = i 4, a 1 a 3, a 2 a 4, and then the condition argmin e C w(e) 2 is equivalent to (3) for Z-freeness. Proof of Theorem 6. We investigate each step in turn. (Step 1). Since the number of defined elements of (h (i,a),(j,b) ) (i,a),(j,b) U is O(n 2 r i=1 d2 i ) = O(n 2 ), we can find an M -convex completion in O(n 2 + n log n) time (recall Remark 4). (Step 2). If we have the value of f(ˆx ), we can compute the value of f(ˆx + χ (i,a) ) in O(r) time since f(ˆx + χ (i,a) ) = f(ˆx ) + c i (a) + h (j,b) supp + (ˆx ) (i,a),(j,b). Hence the time complexity of Step 3 is O(nr 2 ) time. (Step 3). Recall the definition of Gˆx,ŷ in (12) (18). We have Eˆx = O(r(n r)) = O(nr), Eŷ = O(n), E + = O(r), and E = O(r). Hence Eˆx,ŷ = O(nr). Furthermore we need to compute l only on Eˆx, since l is equal to zero on other arcs. If we have the value of f(ˆx ) at hand, we can compute the value of f(ˆx + χ (j,b) χ (i,a) ) in O(r) time since f(ˆx + χ (j,b) χ (i,a) ) = f(ˆx ) c i (a) + (k,c) supp + (ˆx ) h (i,a),(k,c) + c j (b) + (k,c) supp + (ˆx χ (i,a) ) h (j,b),(k,c). Therefore we can construct the auxiliary graph Gˆx,ŷ in O(nr2 ) time. The modified arc length l p is nonnegative [9, Section 5.2]. Hence we can compute p(v) for v V and a shortest path P in Step 3-2 in O(nr + n log n) time by using Dijkstra s algorithm with Fibonacci heaps [5] (see also [17, Section 7.4]). We can update ˆx, ŷ, and p in Step 3-3 in 10

11 O(nr) time. By one iteration of Step 3, the value of ˆx ŷ 1 is decreased by two. Hence the number of iterations of Step 3 is bounded by O(r). Therefore the time complexity of Step 3 is O(nr 3 + nr log n). By the above argument, we see that the proposed algorithm runs in O(nr 3 + nr log n + n 2 ) time. Acknowledgments We thank Kazutoshi Ando and Takanori Maehara for information on the paper [4] in Remark 4. We also thank the referees for helpful comments. This research was initiated at the Trimester Program Combinatorial Optimization at Hausdorff Institute of Mathematics, The first author s research was supported by JSPS Research Fellowship for Young Scientists. The second author s research was supported by The Mitsubishi Foundation, CREST, JST, and JSPS KAK- ENHI Grant Number The last author s research was supported by a Royal Society University Research Fellowship. This project has received funding from the European Research Council (ERC) under the European Union s Horizon 2020 research and innovation programme (grant agreement No ). The paper reflects only the authors views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. References [1] J. L. Bruno, E. G. Coffman Jr., and R. Sethi. Scheduling independent tasks to reduce mean finishing time. Communications of the ACM, 17(7): , [2] M. C. Cooper and S. Živný. Hybrid tractability of valued constraint problems. Artificial Intelligence, 175: , [3] M. C. Cooper and S. Živný. Tractable triangles and cross-free convexity in discrete optimisation. Journal of Artificial Intelligence Research, 44: , [4] M. Farach, S. Kannan, and T. Warnow. A robust model for finding optimal evolutionary trees. Algorithmica, 13: , [5] M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the Association for Computing Machinery, 34: , [6] M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1: , [7] H. Hirai and K. Murota. M-convex functions and tree metrics. Japan Journal of Industrial and Applied Mathematics, 21: , [8] W. A. Horn. Minimizing average flow time with parallel machines. Operations Research, 21(3): , [9] K. Murota. Matrices and Matroids for Systems Analysis. Springer, Heidelberg, [10] K. Murota. Discrete Convex Analysis. SIAM, Philadelphia,

12 [11] K. Murota. Recent developments in discrete convex analysis. In W. Cook, L. Lovász, and J. Vygen, editors, Research Trends in Combinatorial Optimization, chapter 11, pages Springer, Heidelberg, [12] K. Murota. Discrete convex analysis: A tool for economics and game theory. Journal of Mechanism and Institution Design, 1(1): , [13] K. Murota and A. Shioura. M-convex function on generalized polymatroid. Mathematics of Operations Research, 24(1):95 105, [14] T. Petit, J. C. Régin, and C. Bessière. Specific filtering algorithms for over-constrained problems. In Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming (CP 01), pages , Heidelberg, Springer. [15] J. C. Régin. A filtering algorithm for constraints of difference in CSPs. In Proceedings of the 12th National Conference on Artificial Intelligence (AAAI 94), volume 1, pages , [16] H. Reijnierse, A. van Gellekom, and J. A. M. Potters. Verifying gross substitutability. Economic Theory, 20: , [17] A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg, [18] A. Shioura and A. Tamura. Gross substitutes condition and discrete concavity for multi-unit valuations: a survey. Journal of the Operations Research Society of Japan, 58(1):61 103, [19] S. Živný. The Complexity of Valued Constraint Satisfaction Problems. Springer, Heidelberg,